Find the derivative of y = sqrt x, from the first principles.

The definition of the derivative is to determine the limit
of the function for a given point. We'll have in this case x ->
0.

We'll put f(x) = y = sqrt
x

f'(x) = lim [f(x) - f(0)]/(x-0), for
x->0

f'(x) = lim (sqrtx -
sqrt0)/x

We'll substitute x by
0:

lim (sqrtx - sqrt0)/x = (sqrt0 - sqrt0)/0 =
0/0

 Since we have an indetermination case, 0/0, we'll
apply L'Hospital rule:

lim f/g = lim
f'/g'

lim sqrtx/x = lim (sqrt
x)'/x'

lim (sqrt x)'/x' = lim
(1/2sqrt x)/1

We'll substitute x by 0 and we'll
get:

lim (1/2sqrt x)/1 =
(1/2sqrt 0)

f'(x) = 1/2 sqrt
x

f'(0) =
1/2